1 Introduction
The development of drug resistance in response to therapy is a major cause of treatment failure worldwide across a variety of diseases, patient populations, and administered drugs. Resistance to a drug can be present prior to treatment or it can emerge during treatment [1, 2, 3] through a variety of mechanisms. Drug resistance evolves dynamically and often nonuniformly: intercellular variability can lead to faster adaptation to environmental pressures [4] and thus promote the rise of resistant subpopulations in a previouslysusceptible population of cells. Tumor heterogeneity is now understood to be a major contributor to drug resistance in cancer [5, 6, 1, 7], and variability in resistance among bacterial cell populations has been tied to treatment failure [8, 9]. In bacterial evolution experiments, the emergence of multiple antibioticresistant cell lines has been observed [10, 11]. Such clonal diversity, even in a majoritysusceptible cell population, can be the harbinger of drug resistance that ultimately leads to treatment failure: improper treatments can exert a selective evolutionary pressure that leads to the elimination of susceptible cell populations but leaves more resistant cells largely unaffected and able to proliferate due to reduced competition with susceptible cells [12].
The emergence and potential rise to dominance of resistant strains are stochastic processes driven by the dynamic and complex interaction between the influences and interplay of natural selection, demographic stochasticity, and environmental fluctuations. A predictive understanding of the likely evolutionary trajectories toward drug resistance is key to developing treatments that can effectively target and suppress resistant cell populations, but a fully predictive understanding of such processes remains a challenge. Control of evolution, however, does not in principle require full predictability or determinism in evolution. In a closedloop setting, system feedback can mitigate the reliance on precise trajectory knowledge, so long as this feedback can be obtained at reasonable time intervals, the uncertainty and stochasticity can be approximated in an informed manner, and the controller is sufficiently robust to changes in system behavior and parameter fluctuations. In heterogeneous cell environments prone to resistance evolution, the development of such a control policy must balance the need for properly and efficiently targeting all cell types that may either exist at the onset of treatment or that may spontaneously emerge during therapy, while maintaining sufficiently low dosing for toxicity minimization.
The focus of this work is on the development of a closedloop dynamic control prototype, CelluDose, for precision dosing that adaptively targets harmful cell populations of variable drug susceptibility and resistance levels based on discretetime feedback on the targeted cell population structure. The time evolution of such systems is influenced by the potential presence, random emergence, and stochastic growth of multiple cell subpopulations, where demographic stochasticity arises from random processes on the singlecell level. Inappropriate dosing can lead to the proliferation of resistant cell populations, but this interplay is subject to significant levels of stochasticity that must be accounted for in the design of the control algorithm. The target goal of CelluDose is to combat the emergence of drug resistance during treatment by sensitively adjusting dosage and/or switching the administered drug in response to observed feedback on changes in the cell population structure while employing minimal overall drug dosage for toxicity reduction.
Significant and variable stochasticity, nonlinearity in the dynamics, and the potentially highdimensional space of cell types pose significant challenges to the application of classical control methods to such systems. This extends beyond the scenarios considered in this paper: work on more complex disease progression and drug response models with anticipated higher predictive power has been on the rise in recent years, such as in systems pharmacology [13, 14], which brings systems biology approaches into integrative quantitative modeling in the drug discovery process. The expectation is that these increases in model complexity will result in more realistic descriptions of the dynamics that would offer significantly improved predictive power. However, as model complexity increases, the problem of disease progression control by way of modelinformed precision dosing [15, 16, 17, 18, 19] becomes increasingly less tractable, necessitating a tradeoff between the ability to develop a controller for the system and the predictive accuracy of the system model [20].
A shift away from modelbased control methods toward approaches that do not or only partly rely on model details will permit greater freedom in model selection and obviate the need for simplifications that may reduce predictability. The application of modelfree reinforcement learning methods to continuous control tasks has seen significant advances in recent years^{1}^{1}1See, e.g., [21, 22] for recent surveys.
and holds promise for the control of systems for which mathematical optimization and control may be intractable, with potential applications in diverse fields that include robotics, mathematical finance, and healthcare. However, for reinforcement learning to become a standard tool in control engineering and related fields more work is needed in the successful application of continuous reinforcement learning control to highly stochastic and realistic environments.
CelluDose combines a modelfree deep reinforcement learning algorithm with model information in developing an adaptive dosing control policy for the elimination of harmful cell populations with heterogeneous drug responses and random yet significant demographic stochastic changes. Beyond presenting this implementation, the aims of this work are additionally to (1) motivate the use of modelfree reinforcement learning control in the development of nextgeneration precision dosing controllers by demonstrating that a reliable and highly responsive dynamic control policy can be obtained for a complex and stochastic biological system that is intractable with traditional control methods, and (2) showcase how model knowledge may be capitalized on in such a scheme while still benefitting from the flexibility of its modelfree learning approach.
2 Background
2.1 Reinforcement learning for continuous control
The aim of a therapeutic agent targeting harmful cell populations is to produce an effect that leads to the elimination of the harmful cell population while minimizing toxicity by avoiding the use of high doses. Given the possibility of emergence of highresistance cell populations, this task involves a potentially complex tradeoff between acceptably low dosing levels, the need to effectively eliminate the targeted cell populations, and a preference for shorter treatment times to reduce patient morbidity and – in infectious diseases – the possibility of further contagion. As a result, the dosing problem can be cast in somewhat different ways that depend on the particular disease conditions and prevailing treatment preferences. For the purposes of this work, a successful treatment is defined as one in which all targeted harmful cell subpopulations were eliminated by some maximal acceptable treatment time. Within this measure of success optimality is defined as lowest expected cumulative dosing over the course of the treatment. A treatment course is designated as a failure if any targeted cells remain after the maximal treatment time has been reached, with the extent of failure dependent on the number of remaining cells.
Modelfree reinforcement learning (RL) approaches aim to learn an optimal decisionmaking policy through an iterative process in which the policy is gradually improved by sequential tuning of parameters either of the policy itself or of a value (e.g. the actionvalue function) indicative of the policy’s optimality. The data on which learning is done is supplied in the form of transitions from a previous state of the environment to the next state that occurs probabilistically (in stochastic environments) when a certain action is chosen. Each such transition is assigned a reward , and reward maximization drives the optimization of the policy. Learning can be done on a stepbystep basis or episodically, where a single episode consists of multiple steps of transitions. In the case of the dosing problem considered here, an episode is a singlepatient full course of therapy. It ends when either the maximal time has been reached or all cells have been eliminated, whichever comes first. Episodes are thus finite but can be long, with drugs administered at discrete time steps over a continuous range of dosages. At each time interval, a decision is made on which drugs at what doses should be administered based on observations of the current state of the disease, defined as the concentrations of the targeted cell types. However, only at the end of each episode does it become clear whether or not the sequence of therapeutic actions was in fact successful. This credit assignment problem [23] frequently plagues episodic tasks and will be addressed in Section 3.2.
The optimal policy for this decisionmaking process needs to provide nexttimestep dosing guidelines under the optimality guidelines described above and based on potential stochastic evolutionary scenarios described by the model in Section 3.1. This system is continuous in its state space (cell population composition), involves timevarying and potentially high stochasticity, can be highdimensional due to the large number of possiblyoccurring mutant cell types, and involves one or multiple controls (drugs) that if administered intravenously (the scenario of interest here) can in principle take a continuous range of values. For reasons of mechanical operability and medical explainability, a deterministic dosing policy is needed in this case.
Deep deterministic policy gradient (DDPG) [24] is an offpolicy actorcritic deterministic policy deep reinforcement learning algorithm suitable for continuous and highdimensional observation and action spaces. Building on the deterministic policy gradient algorithm [25]
, DDPG employs neural network function approximation through the incorporation of two improvements used in Deep Q Networks
[26, 27] for increasing learning stability: the use of target networks and a replay buffer from which minibatches of transitions are sampled during training. We note that while the action space in the dosing problem is typically lowdimensional (only a few drugs are typically considered for treatment of a particular disease), the observation space (number of cell types) can be quite large, especially if finer observations are available on cellular heterogeneity (finer differentiations in dose responses). For these reasons, the dosing control scheme described here employs the DDPG algorithm.2.2 Modelinformed treatment planning with reinforcement learning
The benefit of a mechanistic modelinformed approach to treatment planning is the ability to explore therapeutic decisions prior to the start of or as an accompaniment to clinical trials, which can then be used to inform clinical trials. The simulationbased use of RL here thus differs from work that employs reinforcement learning for making decisions based on patient data and clinical outcomes, where exploration of treatment parameter space is constrained by data available from previouslyattempted treatments. When this data becomes available during and after clinical trials, these approaches and the approach presented here could be used in a complementary manner to inform and improve individualized treatments.
The use of RL for mechanistic model control in treatment planning is less common than datadriven approaches. We note below a few applications of interest. Qlearning [28], a discrete state and action space RL algorithm, was employed in [29] for closedloop control of propofol anesthesia and in [30]
for control of cancer chemotherapy based on an ordinary differential equations (ODE)based deterministic model of tumor growth. In
[31] a natural actorcritic algorithm [32] was employed for cancer chemotherapy control of a similar ODEbased model, with a binary action space (drug or no drug). Tumors were taken to be uniform and resistance to treatment was not considered as part of the models used in [30] and [31]. In [33] a DDPGbased algorithm was applied to an agentbased model of sepsis progression for obtaining an adaptive cytokine therapy policy.Interest in adaptive control of therapy that takes resistance evolution into account has been on the rise in recent years (see, e.g., [34, 35]) but these efforts have been restricted to simple models that are typically low dimensional and/or deterministic. The approach presented here provides a mechanism for determining and automating dosing in a responsive and robust way that can be generalized to arbitrarily heterogeneous cell populations exhibiting complex and realistic dynamics. Both singledrug and combination therapy control policies are developed in this context, and observations on population composition are supplied at discrete time intervals, as would be expected in a laboratory or clinical scenario. By extension, this work seeks to motivate efforts in the highresolution tracking of celllevel drug resistance progression within an individual patient. From a control engineering perspective, the incorporation of model information into modelfree learning presented here may be usefully transported and adapted to the control of other stochastic systems with an equationsbased description for improved RL learning stability and convergence. For reinforcement learning insights, a detailed analysis is included of policy features that were observed in training.
2.3 Implementation and workflow
Training and implementation rely on three key components: model and simulation development, selection of the subset of drugs of interest, knowledge of the range of likely resistance levels that have been recorded in response to these drugs, and access to diagnostics that can provide temporal data on cell population structure. In this paper, all data supplied in training is simulated; we describe below how such data may be obtained for future proofofconcept validation.
For demonstrative purposes and facility of nearfuture validation, the focus of this work is on drug resistance in evolving bacterial populations subjected to growthinhibiting antibiotic drugs. We note that evolutionary models of cell population dynamics are widely used to understand and model cancer progression and the development of resistance to chemotherapy treatment, and that another promising application area for a CelluDosebased platform is in cancer chemotherapy control.
Future clinical implementation of a CelluDosebased controller could be as an integrated system with the ability to track and supply the trained control model with sensitive measurements of the presence of small populations of cells and equipped with an automated intravenous infusion mechanism. This implementation would be particularly useful in intensive care units (ICUs), where the development of resistance to therapy is a major concern, therapy is performed in the controlled environment of the clinic, and substantial genetic diversity in bacterial strains has been observed [37]. Alternatively, in longerterm therapies, it could be implemented as a decision support tool providing informed dosing recommendations to clinicians. In both cases, observations at the chosen fixed time intervals for which training was done would be supplied in the form of the respective targeted cell type concentrations, where a distinct cell type is (phenotypically) defined by nonnegligible differences in the dose response across the possible ranges of dosages that are permitted for administration.
2.3.1 Modeling and simulation
Although training implements a modelfree reinforcement learning algorithm, model knowledge is used in two ways: to provide the training data (as simulation data) and for algorithm design via feature engineering and reward assignment. This feature engineering is made possible by trajectory estimation from the equationsbased component of the growthmutation model implemented here. This model combines (1) a stochastic differential equationsbased system for approximating the growth of existing cell subpopulations with (2) external random events that perturb this system by creating new subpopulations of resistant cells. These events, which increase the effective dimensionality and alter the composition of the system at random points in time, can be thought of as either
de novo resistant mutations that occur during therapy or as the abovedetectionthreshold rise of mutant populations. The resulting system exhibits variable and substantial levels of stochasticity that can significantly alter its trajectory and potentially lead to treatment failure (see Fig. 1). Since, as shown in Section 4, the learned policies are highly robust to changes in the key mutation parameters, detailed knowledge of mutation rates is not an essential training parameter.Stochasticity can arise from both demographic and environmental fluctuations. The focus here is on demographic stochasticity arising from singlecell variability in birth, death, and mutation processes. For simplicity, spatial homogeneity and a constant environment (other than changes in drug administration) will be assumed here. To obtain a quantifiable estimate of demographic stochasticity, a stochastic physics approach was employed here (Appendix A) in deriving the system of stochastic differential equations describing the heterogeneous cell population time evolution. This permits an informed estimate of the extent of demographic noise from individual cells’ probabilistic behavior and a subsequent quantification of an important contribution to risk in disease progression.
The universal parameters that enter the model are the drugs that may be used with preference information (e.g. which ones are firstline vs. lastline), the appropriate discrete time intervals in which observations become available and dosing can be altered, and the likely spectrum of dose responses. In laboratory evolution experiments [38] this spectrum (e.g. see Fig. 2) can be obtained by allowing the bacteria to naturally evolve under different drug levels and identifying subsequent cell lines accompanied by a characterization of their dose responses through parameter fitting of their growth curves at different drug concentrations. We note that it is not in general necessary to identify all possible and potential mutations prior to training. In particular, knowledge of all possible genetic changes is not in principle necessary: it suffices to characterize the expected relevant diversity in drug response^{2}^{2}2Such studies, however, are frequently accompanied by genotype analysis.. The idea is to provide enough training data for the model so that new variants are unlikely to exhibit dose responses considerably different from all those already identified. During an experimental run of the trained RL model, when a newlyemergent cell type subpopulation is detected it can thus be binned without significant loss of accuracy into the statespace dimension (Section 3.2) of a previouslyidentified variant.
2.3.2 Observations and diagnostics
Early, fast, and highresolution detection of heterogeneity in drug response is crucial for identifying resistant cell subpopulations before such populations spread [39, 40, 8]. Resistance detection may be done through phenotypebased drug susceptibility testing or by capitalizing on advances in genomic sequencing in combination with predictions of resistance from genotype [11, 41]. A key aspect is the requisite ability to effectively detect low concentrations of resistant cells in an otherwise suseptible population. Singlecell analysis is showing mounting promise in this regard and is being used to resolve heterogeneity in tumors [42] and in bacterial populations [43, 44]. Direct phenotypebased detection of small resistant bacterial subpopulations in proportions as low as of the total population was performed in [45]. For genotype detection, singlecell sequencing [46] can be combined with predictions of resistance from genotype for highresolution detection of a population’s resistance profile.
These advances could be applied to nearfuture laboratory validation of an in vitro CelluDose controller via combination with software implementation into an automated liquid handling system in bacterial evolution experiments. Greater automation, efficiency, and standardization are nonetheless needed for these technologies to be routinely employed as part of a clinical controller apparatus, and, in part, the intention of this work is to motivate efforts in the clinical development and use of singlecell analysis for resistance profiling by demonstrating its potential utility for automated disease control.
3 Methods
3.1 Stochastic modeling and simulation of cell population evolution
The evolutionary fate of an individual cell during any given time interval depends on the timing of its cell cycle and its interaction with and response to its environment. The model of evolutionary dynamics employed here depends on three processes: cell birth, cell death, and inheritable changes in cell characteristics that lead to observable changes in growth under inhibition by an antibiotic. We consider here drugs that suppress cell growth by inhibiting processes essential for cell division by binding to relevant targets and thus leading to reduced cell birth rates. The doseresponse relationship can be described by a Hilltype relationship [47, 48]. Here a modified Hilltype equation will be employed that includes the potential use of multiple drugs with concentrations given by . The growth rate of a particular cell type as a function of the drug concentrations to which cells are exposed is thus taken to be
(1) 
where is the rate of cell birth, is its drugfree birth rate, is the rate of cell death, and describes the extent of resistance of cell type to drug . For simplicity, since drugs are assumed here to affect only birth rates, changes that confer resistance are assumed to affect only rather than . Resources for cell growth are assumed to be limited, with the environmental carrying capacity denoted by , indicating the total bacterial cell population size beyond which no further population increases take place due to resource saturation.
Evolution is an inherently stochastic process that depends on processes, as noted above, that individual cells undergo. However, individual (agent)based simulations are generally very costly for larger numbers of cells and, moreover, obscure insight into any approximate deterministic information about the system. At very high population levels (that can be treated as effectively infinite) a deterministic description of system dynamics is appropriate. In the dynamic growth scenario considered here, large size discrepancies can exist at any time between the various concurrent cell populations, and a single population can decay or proliferate as the simulation progresses. The extent of demographic stochasticity in such systems will thus change in the course of evolution. For efficient modeling at higher population levels it is imperative to use a simulation model whose runtime does not scale with population size, but for accurate modeling a proper estimate of stochasticity must be included. This is done here by applying methods from stochastic physics [49, 50, 51]
to derive a diffusion approximation of a master equation describing the continuoustime evolution of the probability distribution of the population being in some state
, where are the (discrete) subpopulation levels for the different cell types. This approximation relies on the assumption that the environmental carrying capacity is large compared to individual subpopulation levels and results in the evolution of the system being described by a system of stochastic differential equations (SDEs) governing the time evolution of the subpopulation levels of the different cell types given by the continuous variables , where is the number of distinct phenotypes that may arise. We only consider phenotypes whose growth rate at nonzero drug concentrations is higher than the baseline susceptible phenotype (wildtype) due to the negative evolutionary selection experienced by phenotypes with lower growth rates. The system evolves according to (Appendix A)From a control standpoint, the advantage of using SDEs over an agentbased model is that we can capitalize on the equationsbased description for trajectory estimation and use that information in feature engineering and reward assignment below. Note that the stochastic noise in these equations is not put in ad hoc but instead emerges naturally from population demographic randomness that arises from demographic processes on the level of single cells^{3}^{3}3Other sources of noise due to environmental factors would in general be expected to exist as well and can be included in the model. Here we restrict to demographic stochasticity.. For more realistic dynamics at very low cell levels, when the level of a subpopulation falls below 10 cells, the number of cells is discretized by a random choice (equal probability) to round up or down to the nearest integer cells after each stochastic simulation step; when cell numbers fall below 1 the subpopulation level is set to zero.
A few comments are due on the large approximation above (see Appendix A for details) in the context of the work here. In general, as , this approximation ceases to provide a good description of the fluctuations in the system. In that regime, the large approximation leads to deterministic dynamics and neglects the effect of population fluctuations near carrying capacity, which are expected in biological systems. This is not a concern in the scenario considered here: since is assumed here to be large, as approaches its dynamics become effectively deterministic and welldescribed by the logistic drift term alone; moreover, the treatment goal is to reduce the cell populations rather than maintain some distribution near carrying capacity, so that after the onset of treatment populations should spend little time in the vicinity of the resource capacity. As a result, any fluctuations at that point will have negligible effect on the progression of treatment. As the population decreases it moves farther away from the resource capacity and into the regime in which the diffusion term is a valid approximation of stochasticity, whose contribution to the dynamics also increases at lower population sizes due to the dependence on . The logistic deterministic dependence, however, is relevant for modeling accuracy at the initial stages of treatment (where it can affect the growth rate) as well as for training: a finite resource capacity introduces into training the issue of low growth despite population proliferation (and hence treatment failure). This needed to be addressed with a particular reward assignment scheme, discussed in Section 3.2, and was done for potential future generalization to scenarios where the population levels may stay for longer closer to the carrying capacity due to treatment constraints. Lastly, setting a carrying capacity motivated a nonarbitrary scale in the problem for feature rescaling (Appendix B).
For clarity, in this paper the term decision time step will be used to refer to the length of time between each dosing decision and thus defines the RL time step, whereas SDE simulation step will be used to refer to a stochastic simulation step. A single decision time step thus contains a large number of SDE simulation steps. The evolution of Eq. (2) was simulated via the EulerMaruyama method with a step size of 0.01 over the 4hour decision time step (unless all populations drop to zero before the end of the decision time step is reached).
Mutations are modeled here as random discrete external events that perturb the system of Eq. 2 by injecting new population members into a particular subpopulation . For simplicity, resistance is assumed to be acquired as a onestep process by mutation from wildtype; while resistance has been observed to increase with the accumulation of mutations, single mutations can suffice to confer resistance [52, 53]. During training (Section 3.2) there is a probability that any mutations will occur during an episode, so that mutations are precluded from occurring in most episodes. This is done in order to permit the policy to identify an optimal baseline (no mutation) dosing schedule.
When mutations are allowed, they are permitted to occur at any time step up until 24 hours before the maximal time allowed for treatment, where the maximal time is set at substantially longer than the time necessary to eliminate the cell population if no mutations were to occur. This restriction is set in place as any mutations at the very end of treatment will always necessitate an extension past the maximal time, which is already built in by setting the maximal time at longer than the necessary time. Within these bounds mutations may occur to any (or all) of a potential set of prespecified phenotypes at any decision time step with some fixed small probability . When a resistant population randomly emerges in this manner, its population size is assigned through a combination of random number generation and size estimation based on the expected deterministic increase in the baseline (bl) population level if it were the only present cell type:
(3) 
where is the decision time interval and is generated randomly (in steps of
) from a uniform distribution between
and , in keeping with typical approximate observed rates of occurrence of resistant mutant cells in bacterial populations [8]). The rationale behind Eq. (3) is that larger populations will naturally give rise to more mutant cells. Rounding up is done both due to the typically low cell numbers (individual statistics modeling, as noted above), and in order to partially offset the low probability of mutation occurrence. If a mutation occurs to a particular phenotype, then is added to that phenotype’s subpopulation and, as an approximation, removed from the baseline phenotype’s subpopulation, as fewer cells are now available for its further growth. These population adjustments are done prior to simulating the stochastic evolution of the system via Eq. (2) for that decision time step.Fig. 1 shows several model simulations for a constant low dose case that is above the susceptibility level of the initiallydominant phenotype but below that of three potential variants. All dose responses are shown in their respective colors in Fig. 2; the (simulated) parameters used are not specific to any particular organism or drug but fall within common concentration and growth rate ranges.
3.2 Learning an adaptive dosing policy
As discussed in Section 2.1, DDPG [24]
was employed for training. Below we describe the state and action spaces and reward assignment. The neural network architecture and hyperparameter choices for the neural network and RL components are given in Appendix
B.In training, the maximal time for an episode was set at 7 days, with dosing decision steps of 4 hours. Although time scales for observations will be technologydependent and possibly longer, this time interval was put in place in order to demonstrate the ability of the algorithm to handle episodes with many decision time steps, and hence a nearcontinuous dose adjustment, and produce a robust control policy able to adaptively and responsively adjust to changes in the cell population composition. To put in clinical context the maximal treatment time chosen, we note that a 2000 study [54] found that in a studied patient cohort typical ICU antibiotic courses lasted for 420 days, with an average length of 10 days; after 7 days of a standard antibiotic course 35% of patients in the study were found to have developed antibiotic resistance (new resistant patterns in the original pathogen) or superinfections (newlydetected pathogenic organisms not originally present). Shorter treatment times (3 days) in that study correlated with a lower rate of antibiotic resistance (15% at 7 days). Stopping antibiotic treatment prematurely, however, runs the risk of failing to eliminate the pathogenic bacteria, suggesting that continuous monitoring and proper dose adjustments – as recommended here – are essential in combatting treatment failure in severe infections.
All code (simulation and RL) was written in Python, with the use of PyTorch
[55]for the deep learning components.
3.2.1 State space: modelinformed feature engineering
At each decision time step an observation of the current composition of the cell population, i.e. types and respective concentrations is made. Cell types that are known to potentially arise but are presently unobserved are assigned . As a result of model knowledge in the form of the SDE system (Eq. 2), however, additional feature combinations of can be used to supplement and improve learning. In particular, the current growth rate of the total cell population is highly indicative of drug effectiveness, but it is typically difficult to measure and necessitates taking multiple observations over some fixed time interval (e.g. 1 hour) prior to the end of a decision time step. Instead, the instantaneous growth rate can be calculated as
(4) 
and approximated directly from observed features, , at time by noting that in the deterministic limit the numerator is simply given by
(5) 
where is fixed and known (Eqn. 1) during any decision time step. Hence is a function exclusively of the action taken (doses chosen) and observations of cell concentrations .
Use of this feature combination, which would not be known in a blackbox simulation, was found to be instrumental in shaping the reward signal at nonterminal time steps and in leading to optimal policy learning, suggesting that such incorporation of information could in general assist in driving policy convergence when model knowledge is available but a modelfree approach is preferred due to the complexity of the dynamics.
3.2.2 Action space
Training was performed with both a single drug and two drugs under the same training hyperparameters (with the exception of drug preference). Drugs were set to take a continuous range of values from zero up to some maximal value , and the action of drugs on the cell populations is expressed through the doseresponse as in Eq. (1) and the SDE system of Eq. (2). This maximal value was used for reward scaling as shown in Section 3.2.3 rather than for placing a hard limit on the amount of drug allowed to be administered. It was set for both drugs at 8 times the highest value in the set for all drugs . This very high value was chosen in order to allow enough initial dosing climb during training; in practice, conservative dosing was implemented through the reward assignment alone.
3.2.3 Reward assignment
While the state space includes information on separate cell subpopulations, the terminal (endofepisode) reward is assigned based on only the total cell population. As noted above, a successful episode concludes in the elimination of the targeted cell population by the maximal allowed treatment time . In training, different drugs are assigned preference through penalty weights , so that use of a lastline drug will incur a higher penalty than that of a firstline drug, . To penalize for higher cumulative drug dosages, the reward at the end of a successful episode is assigned as
(6) 
where is the dosage of drug administered at the th decision time step, are constants, and is the length of each decision time step. When the episode fails (the cell population is above zero by ) a negative reward is assigned with an additional penalty that increases with a higher ratio of remaining cell concentrations to the initial cell population in order to guide learning:
(7) 
where . Since the episodes can be long, in order to address the credit assignment problem [23] a guiding signal was added at each time step in the form of potentialbased reward shaping [56]. Policy optimality was proved to be invariant under this type of reward signal, and it can thus provide a crucial learning signal in episodic tasks. It is given by
(8) 
where is the RL discount factor, is the potential function, and is the state space. Since the drugs are assumed here to affect cell mechanisms responsible for cell birth, population growth provides a direct indicator of the efficacy of the drug. The potential function was therefore set here to
where , is given by Eq. 4, and is the zerodrug fullpopulation deterministic growth rate at (see also Eq. 5):
(9) 
Although incentives for low dosing were built into the terminal reward (Eq. 6), for guiding training toward low dosing it was necessary to provide some incentive for low dosing at each decision time step. The step reward was therefore assigned as the sum of the shaping reward and a dosage penalty:
(10) 
where is the specific penalty for each drug, and is a binaryvalued function explained below. For a previous implementation of stepwise action penalties with potentialbased reward shaping see [33].
An issue peculiar to the problem of a population growing under limited resources is that if insufficient dosing is applied and the population subsequently quickly grows and reaches its carrying capacity, very little change in growth (or population) will occur yet the state of infection will be at its maximal worst state. However, the shaping reward (8) – which is based on changes from one decision time step to the next – will provide insufficient penalty for this (with the penalty arising solely from the weighing). On the other hand, as a result of the inhibitor penalty in (10), the algorithm may continue to reduce the dosing even further rather than increasing it to escape this suboptimal policy. The result is that the targeted cell population proliferates at maximal capacity while dosing fluctuates around the minimal value of zero drug administration. This zerodosage convergence problem was successfully resolved by assigning a significanty lower (20fold) weight to the midepisode dosage penalty if the total cell population exceeded the initial cell population by more than 1 cell for every initially present cells. The incorporation of the binary penalty function led to significantly improved stability and reduced sensitivity to the exact choice of .
4 Training and results
Up to four cell types with different dose responses were permitted to occur in each training scenario, with the most susceptible type initially dominating the population but with different cell populations allowed to emerge as early as the first time step. For singledrug training the dose responses shown in Fig. 2 were assumed. For combination therapy (twodrug) training, one drug was defined as the “firstline” drug and the second as the “lastline” drug via respective choices of . The responses of the black and blue phenotypes to both the firstline and lastline drugs were taken to be identical to those shown in Fig. 2. The more resistant phenotypes (red and green) were assumed to be completely resistant to the firstline drug (by assigning them a very large value of ) and to have the dose response shown in Fig. 2 to the lastresort drug.
In the singledrug case, in the early stages of training the dosages were increased rapidly and uniformly in all time steps; at that point the dosages were too high for any failures to occur. After about 180 episodes considerable decreases in dosing began to occur (also uniformly). The rate of decrease slowed down once the dosage reached the approximate level necessary to eliminate the most resistant cell subpopulations (at about 500 epidoes), with dosing still uniform over the entire course of treatment. At that point increasing specialization in dosage at different time intervals in response to different mutations began to occur, and the baseline (nomutation) dosing continued to further decrease in decision time steps following the first time step. Dosing during the first time step remained high even as subsequent doses were brought substantially lower. For these doses training gradually alternated for a long time between a gradual monotonic decrease, uniform dosing, and a gradual monotonic increase, eventually establishing the monotonic increase characteristic of the baseline dosing observed after training converged (top row of Fig. 4). Dosing during the first time step was eventually brought further down. In the early stage of training episode failures would occasionally occur; this ceased to happen later in training, at which point further improvements in learning were on dosing optimization.
In dualdrug training, similar uniform increases were observed in early training for both drugs, with the firstline drug (lower penalty) initially increasing more, but with this increase reversed after about 100 episodes (recall that lower dosages of the firstline drug are needed to suppress the mutations that are not completely resistant to it). By episodes into training the firstline drug was not administered at all  but treatments were still successful since the lastline drug is capable of suppressing all cell types, albeit at a higher penalty. The lastline dosage continued to be uniformly decreased until treatment failures started occurring and was subsequently increased to the level at which it could suppress mutant populations without specialization. Decreases again began occurring at that point, and the policy also started applying the firstline drug again. Training from that point on involved specialization in response to mutation and experimentation with various amounts of the firstline and lastline drugs.
Training was stopped after no further reduction was observed in the MSE loss and the actor policy loss had stabilized. The MSE loss flattened at , and this was observed to occur in singledrug training after a somewhat smaller number of training episodes than in dualdrug training for the same parameters (Fig. 3). The policy parameters were saved every 100 episodes for later reference and the resulting policies were analyzed after training. Following training convergence (determined as described above) dosing was generally similar for policy parameters recorded at different episodes, with some minor fluctuations observed. In the dualdrug case, when no mutations were experienced, the learned policies either administered no lastline drug at all or negligibly small amounts of it only in the first and last time step (Fig 5).
Responsive adaptation to random events and system perturbations
A hallmark of both the singledrug and combination therapy policies is the responsive adaptation to random population composition changes (Fig. 4 and Fig. 5) insofar as both the appearance of a new cell type and stochastic fluctuations of alreadypresent cell types. On a test set of 1000 episodes simulated randomly with training parameters a 100% success rate was obtained in both the singledrug and dualdrug cases (all episodes recorded involved the occurrence of mutations). Multiple concurrent subpopulations are handled well by the policies learned. In singledrug therapy, higher drug pulses are only administered after observing an emergent resistant subpopulation, and baseline dosing is restored following the elimination of this subpopulation (Fig 4). The dosage administered in these pulses directly increases with the resistance and population of the observed cell type. In combination therapy (Fig 5), the policy switches to the lastline drug when the red and green phenotypes are observed and switches back to the firstline drug after these cells are eliminated. For the lowresistance blue phenotype appropriate dosing increases are done exclusively with the firstline drug.
This is particularly remarkable given that the black and blue phenotypes are susceptible to the same extent to both the first and lastline drugs. The sharp difference in drug administration is thus attributable to the differing incentives supplied in the reward assignment ().
Robustness to changes in mutation parameters
In a given episode, the rate of resistant subpopulation emergence and the extent of it were controlled by and , respectively. To assess the robustness of the policy to variations in these parameters the success rates of policies shown in Fig. 4 (one drug) and Fig. 5 (two drugs) were tested over parameter combinations (100 trials for each combination) that significantly exceeded the training parameters. No deterioration in performance was found for parameter combinations within and substantially above the expected biological values: 100% success was recorded for testing of up to a 10fold increase in and more than a 1000fold increase in , which controls the number of resistant mutants observed. No failures were recorded for of up to of the expected increase in susceptible cells in any time interval (recall that the actual number of cells is randomly generated according to Eq. 3 with respect to ). Slight performance deterioration was found at the very high end of the range tested (), but success rates of were still recorded for these parameter combinations^{4}^{4}4Tests were performed with the policy parameters obtained after 95,000 episodes of training for both the singledrug and dual drug policies, shown in Fig. 3.; we note that this high end of the range is in significant excess of resistant cell proportions that may be naturally expected [8].
Generally, the robustness of the learned policies to such strong parameter deviations implies that it may be possible to apply them to cases where the detection threshold for newly emergent populations is higher due to lowerresolution diagnostics. However, modifications accounting for the resolution at subsequent time steps will likely be needed in such cases.
Policy behavior in the administration of baseline dosing
Typical singledrug baseline (nomutation episode) dosing exhibits slightly elevated dosages at the start of each episode that then mildly drop. Subsequent decision time steps involve approximatelyconstant but mildly monotonically increasing dosages. An episode can terminate at various points in time even in the absence of mutations (top rows of Figs. 4 and 5) due to variations in the initial cell population size of up to an order of magnitude as well as demographic fluctuations at low populations toward the end of treatment. The dosing is seen to adjust in accordance with these variations.
We can gain insight into the reason for this mild monotonic increase by comparing the singledrug dosing with the dualdrug dosing policies, where we can isolate the effects of two of the mutations due to the differing responses to the two drugs. In the combination therapy case, after the first time step the firstline drug is administered at a nearlyconstant value with no increase in dosing as the treatment proceeds. On the other hand, as noted above, occasional minor lastline drug dosages at the very beginning and end of nonmutation episodes can be observed in some runs^{5}^{5}5At the beginning of the treatment this was only observed to coincide with a slight dip in the firstline dosage.. One explanation for these dosages is that they may arise from learning that if resistant phenotypes emerge at the beginning or the very end of treatment then an immediate response via lastline dosing is paramount to achieving an optimal treatment not involving higherthannecessary dosing. In the singledrug case a dosage increase can combat these mutations, and the observed monotonic rise may thus emerge from the policy’s anticipation of more resistant phenotypes. It is therefore appears plausible that the anticipation of latestage higherresistance mutations drives this increase rather than any aspects of the susceptible cell population growth.
Learned preference for short treatment times
All explicit optimization incentives that were given involved the dosage administered rather than the time of treatment (Section 3.2). As previously explained, the main challenge in the dosing problem is the balancing of the higher dosages needed to suppress resistant cell populations with the need to keep toxicity low. Rewards in the setup implemented here purposely did not incorporate any direct incentives for shorter treatment times, focusing instead on dosage minimization. Although conservative dosing was a strong feature of all learned policies, the policies learned in training involved a strong preference for shorter treatment times even at the expense of higher dosing, typically eliminating the cell population well in advance of the maximal treatment time. This is particularly evident from a comparison of the policy response in its dosing vs. its treatment time (Figs. 5(a) and 5(b)) to increases in the mutation rate per step () and resistant cell supply via (Eq. 3): the policy compensates for the increased resistance almost exclusively through dosage increases, while treatment time is kept nearly constant. This behavior may arise through a combination of learned preferences based on dosing penalties and mutation parameters, and in training for a real clinical scenario a “clinicianintheloop” approach would permit informed choices in this regard.
5 Discussion
This paper presented a method for singledrug and combination therapy feedback control that results in policies capable of responsive and robust adaptation to changes in a stochastic dynamical system of evolving cell populations. Modelfree deep reinforcement learning was supplemented with model information on system trajectory estimation into feature engineering and reward assignment, which was found to significantly improve learning. Various aspects of the resulting policies were investigated in this study and may help guide future efforts training similar systems in which some trajectory estimation is possible, the probability of random perturbing events may not be known, and a tradeoff between conflicting target goals (here: low toxicity with the need to properly target resistant cell populations) exists.
Given the demonstrated ability of the policies that were learned in robustly adjusting to large variations in certain parameters on which they were not trained, an interesting question for future exploration is how accurately the underlying structure of the model – here logistic growth dynamics – needs to be known for policy training. Higher flexibility in this regard would enable controller development with more limited predictive knowledge of the dynamics. The resolution of feedback observations and especially their frequency is likely to play an important role in this flexibility. Here, observations were assumed to be generated at discrete time points but be reflective of the population composition at these time points. Practical implementation would likely necessitate accounting for a certain time delay in obtaining such observations, and future directions for the development of CelluDose include the incorporation of delayed feedback and lowerresolution observations that lead to partial observability of the system.
As higherresolution diagnostics, more sophisticated modeling techniques, and a recognition of the need for patientspecific care are making increasing gains in healthcare, the need for appropriate therapeutic control methods is likely to rise. The ultimate goal in precision dosing – a feedback loop in which drug absorption and response are tightly monitored and used to determine subsequent dosing – remains for now an elusive goal in all but a few cases [57]. The method presented here is intended for eventual use in this context under consistent monitoring of the targeted cell population composition in terms of the variability in its dose responses.
An important advantage of modeling and simulation in developing drug dosage controllers is the ability to investigate a large range of drug schedules even prior to clinical trials and to thus potentially inform the development of clinical trials. The focus of the implementation presented here is on the drug responses and trajectories of an evolving population of cells. For clinical applications, modeling would need to additionally include patientrelevant models of drug absorption and immune system response. With appropriate diagnostics, the setup described here could be implemented for the automated control of laboratory bacterial evolution experiments and establish the roles and interplay of decision time intervals, diagnostic resolution, and model uncertainty as part of the controller design. This would be a crucial step toward future clinical application and a validation of the utility of artificial intelligencedriven approaches in simulationbased biomedical controller design.
Acknowledgements
I am grateful to Tommaso Biancalani, Roger Brockett, and Finale DoshiVelez for useful discussions. This work was supported by NIH grant 5R01GM12404402 to Eugene Shakhnovich.
Appendix A Master equation to stochastic differential equations
a.1 Master equation for the population dynamics of phenotypes
Let denote a single cell of phenotype . We consider phenotype with equal resource utilization evolving subject to a resource capacity given by . Following a simplified setup of [58], we initially consider a spatial grid permitting up to one individual of type per site. If empty, individual in site is denoted by . The rates of birth and death of cells of type are denoted by and respectively:
Birth:  
Death: 
At each time step we sample the population. On a fraction of these events we randomly choose two individuals and allow them to interact, but since we do not consider direct competition effects other than through shared limited resources, we ignore combinations in which both cells were picked and therefore restrict to picking only the combination and . In a fraction of these events we choose only one individual randomly (if only ’s are drawn, the drawing is done again). The probabilities of picking the different combinations are given by
where is the population of phenotype . We will denote the transition probabilities from state to state by . They are thus given by
where for simplicity we replace as we will be considering the large limit for the subsequent diffusion approximation. The master equation is given by
(11)  
subject to the initial condition (Kronecker delta, not to be confused with the death rates ). We must also impose boundary conditions,
a.2 Diffusion approximation: the FokkerPlanck equation and Ito stochastic differential equations
By assuming that the resource capacity is large compared to the population levels , we can approximate as continuous variables; we denote and define
where we have rescaled the death and birth rates as and . Eqn. (11) can then be rewritten as
Taylor expanding around to second order [51] we have that
which yields, after rescaling time as , the FokkerPlanck equation
which corresponds to the system of Ito stochastic differential equations
where the white noise satisfies
(12) 
and is the Kronecker delta (not to be confused with the death rate ). We have that
If we redfine as the levels (concentrations) rather than proportions of the populations, we observe that the fraction
approaches 1 in the large limit, in which case we have
yielding the system (2)
for . To prevent the term in the square root of the noise from occasionally briefly dipping below zero and resulting in imaginary numbers (due to the cell subpopulations potentially dipping below zero during a simulation step before being set to zero), this term is clipped at zero.
Appendix B Neural network architecture and hyperparamer choices
The actor and critic network (and respective target network) architecture used is the same as in the original DDPG paper [24], with the notable difference that significant performance improvement was obtained with the addition of an extra hidden layer (45 units) in the actor network^{6}^{6}6Notably, the addition of an extra layer in the critic network led to degraded performance.
and that narrower hidden layers (60 and 45 units respectively) were implemented to avoid overfitting. All hidden layers employed rectified linear activation functions; the output layer of the actor was set to a hard tanh with range between 0 and a chosen maximum dose in order to allow for the dosage to drop to exactly zero (particularly important when multiple drugs are considered). Weights were initialized randomly but biases were set to a positive value informed by the evolution of the deterministic part of (
2) (see Appendix C for a detailed description). This was found to prevent convergence to a suboptimal policy in which the minimal control (0) was applied at every time step early in the training.No batch normalization was used at hidden layers, but features were rescaled prior to being added to the replay buffer in the following manner. Cell concentrations
were rescaled asSince cell concentrations can vary by multiple orders of magnitude, this was found to be necessary for smooth convergence. The growth rate was rescaled as
where was defined in (9) and was set to the negative of the death rate, , which is the lowest growth rate ( is the highest rate of decline) possible in the system.
Adam optimization was used with learning rates of for the actor network and for the critic network as learning was found to be unstable for higher rates. Training was done with minibatch size of 128 and a replay buffer of size . Soft updates to the target networks were done with and the RL discount factor was set at . Exploration was done with an OrnsteinUhlenbeck process with , , and . Parameters used in the reward choice that resulted in successful training were , , , , . Singledrug training was done with both and . Combination therapy (two drugs) training was done with under the assumption that the drug able to effectively target the higherresistance cells is a lastresort drug that also involves higher toxicity.
Appendix C Bias initialization
If we assume the lowestvariance policy – constant uniform dosing – and permit no mutations to occur, then in deterministic evolution the number of baselinephenotype cells under constant dose
starting from a population will be given at time by the solution of the noisefree system equivalent of (2):(13) 
where . If the population must be reduced to (here, cell) within a time , then the dosage must be no less than
for treatment to be successful. Initial cell populations were allowed to vary by up to an order of magnitude; by taking the maximum value allowed in this range we compute and set the bias to five times this value or to 75% of the maximal inhibit  whichever is smallest (typically ). was set to just under 1 cell (0.98) as the population falling below a single cell marked the conclusion of a successful episode. The full initialization for both actor and critic networks were saved and used in subsequent runs for training consistency.
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